Noether’s Theorem - Physics Courses
Chapter 7Noether s Continuous Symmetry Implies Conserved ChargesConsider a particle moving in two dimensions under the influence of an external potentialU(r). The potential is a function only of the magnitude of the vectorr. The Lagrangian isthenL=T U=12m r2+r2 2 U(r),( )where we have chosen generalized coordinates (r, ). The momentum conjugate to isp =mr2 . The generalized forceF clearly vanishes, sinceLdoes not depend on thecoordinate . (One says thatLis cyclic in .) Thus, althoughr=r(t) and = (t)will in general be time-dependent, the combinationp =mr2 is constant. This is theconserved angular momentum about the instead the particle moved in a potentialU(y), independent ofx, then writingL=12m x2+ y2 U(y),( )we have that the momentumpx= L/ x=m xis conserved, because the generalized forceFx= L/ x= 0 vanishes.
Noether’s Theorem 7.1 Continuous Symmetry Implies Conserved Charges Consider a particle moving in two dimensions under the influence of an external potential U(r). The potential is a function only of the magnitude of the vector r. The Lagrangian is then L= T−U= 1 2m r˙2 +r2 φ˙2 −U(r) , (7.1) where we have chosen generalized ...
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